In One Flew Over the Cuckoo's Nest by Ken Kesey, what does freedom mean to the patients? What does it mean to Nurse Ratchett?

One important quotation from the novel is:

“If you don't watch it people will force you one way or the other, into doing what they think you should do, or into just being mule-stubborn and doing the opposite out of spite.”

This encapsulates the idea of freedom for the patients, namely freedom from external constraints and the ability to live their lives completely as they choose. The vision Kesey develops is Utopian and anarchic, very much part of the Beat anti-Establishment ethos. It suggests not just freedom from external authority or compulsion, but also freedom from conventional ideas that restrict the scope of individual action. At times, though, the gestures of freedom such as buying sex from prostitutes, getting drunk, indulging in recreational drugs, and engaging in pointless violence seem less admirable than Kesey intends them to be. Kesey finds the standard notion of "sanity" itself oppressive. 
Nurse Ratched appears intended as the villain of the novel, representing the forces of order and oppression within the hospital with the protagonist McMurphy constantly attempting to rebel against her and humiliate her. Her notion of freedom is more concerned with order. In other words, a free society for her is one in which people fulfill their responsibilities and respect authority, and thus one in which one is free from threats of violence.